Existence of Unique Fixed-Point Results in Fuzzy Metric Spaces and Its Utilization

Ajay Soni; Pravin Kumar

doi:10.69980/ajpr.v28i5.399

Authors

Ajay Soni
Pravin Kumar

DOI:

https://doi.org/10.69980/ajpr.v28i5.399

Keywords:

Fuzzy metric spaces, fixed-point theory, unique fixed-point results, utilization, fuzzy mathematics.

Abstract

The study of fixed-point theorems within fuzzy metric spaces is pivotal due to its extensive applications across various scientific and engineering disciplines. This paper investigates the existence and uniqueness of fixed-point results in fuzzy metric spaces, addressing a significant gap in current mathematical literature. We present several new theorems that establish conditions under which fixed points exist and are unique, thereby extending and generalizing existing results. Utilizing rigorous mathematical analysis and advanced techniques in fuzzy mathematics, we provide comprehensive proofs to support our findings. The significance of these results lies in their potential applications, which span areas such as systems engineering, computer algorithms, and economic modeling. By demonstrating how these theorems can be applied in practical scenarios, we highlight their utility in solving real-world problems, thereby underscoring the broader impact of our work.

Author Biographies

Ajay Soni

Department of Mathematics, IES College of Technology, Bhopal, India.

Pravin Kumar

Department of Basic Science, IES University Bhopal, India.

References

1. Shamas, I., Rehman, S. U., Aydi, H., Mahmood, T., & Ameer, E. (2021). Unique Fixed‐Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations. Journal of Function Spaces, 2021(1), 4429173.

2. Sonam, Bhardwaj, R., & Narayan, S. (2023). Fixed point results in soft fuzzy metric spaces. Mathematics, 11(14), 3189.

3. Furqan, Salman, Hüseyin Işık, and Naeem Saleem. "Fuzzy triple controlled metric spaces and related fixed point results." Journal of Function Spaces 2021.1 (2021): 9936992.

4. Konwar, Nabanita. "Extension of fixed point results in intuitionistic fuzzy b metric space." Journal of Intelligent & Fuzzy Systems 39.5 (2020): 7831-7841.

5. Kanwal, Shazia, et al. "Existence of Fixed Points in Fuzzy Strong b‐Metric Spaces." Mathematical Problems in Engineering 2022.1 (2022): 2582192.

6. Romaguera, S., & Tirado, P. (2020). Characterizing complete fuzzy metric spaces via fixed point results. Mathematics, 8(2), 273.

7. Turkoglu, D., Alaca, C. İ. H. A. N. G. İ. R., Cho, Y. J., & Yildiz, C. (2006). Common fixed point theorems in intuitionistic fuzzy metric spaces. Journal of applied mathematics and computing, 22, 411-424.

8. Ashraf, M. S., Ali, R., & Hussain, N. (2020). New fuzzy fixed point results in generalized fuzzy metric spaces with application to integral equations. IEEE Access, 8, 91653-91660.

9. Ješić, S. N., & Babačev, N. A. (2008). Common fixed point theorems in intuitionistic fuzzy metric spaces and L-fuzzy metric spaces with nonlinear contractive condition. Chaos, Solitons & Fractals, 37(3), 675-687.

10. Hussain, N., Latif, A., & Iqbal, I. (2015). Fixed point results for generalized F-contractions in modular metric and fuzzy metric spaces. Fixed Point Theory and Applications, 2015, 1-20.

11. Beg, I., Sedghi, S., & Shobe, N. (2013). Fixed point theorems in fuzzy metric spaces. International Journal of Analysis, 2013.

12. Sonam, Rathore, V., Pal, A., Bhardwaj, R., & Narayan, S. (2023). Fixed‐Point Results for Mappings Satisfying Implicit Relation in Orthogonal Fuzzy Metric Spaces. Advances in Fuzzy Systems, 2023(1), 5037401.

13. Gupta, Vishal, and Ashima Kanwar. "V-fuzzy metric space and related fixed point theorems." Fixed point theory and applications 2016 (2016): 1-17.

14. Sharma, P. K. (2020). Some common fixed point theorems for sequence of self mappings in fuzzy metric space with property (CLRg). J. Math. Comput. Sci., 10(5), 1499-1509.

15. Rao, K. P. R., Altun, I., & Bindu, S. H. (2011). Common Coupled Fixed‐Point Theorems in Generalized Fuzzy Metric Spaces. Advances in fuzzy systems, 2011(1), 986748.

16. Kumar, S., Vats, R. K., Singh, V., & Garg, S. K. (2010). Some Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces. Int. J. Math. Anal, 4(26), 1255-1270.

17. Gregori, V., Miñana, J. J., & Miravet, D. (2019). Extended fuzzy metrics and fixed point theorems. Mathematics, 7(3), 303.

18. Jung, J. S., Cho, Y. J., & Kim, J. K. (1994). Minimization theorems for fixed point theorems in fuzzy metric spaces and applications. Fuzzy sets and systems, 61(2), 199-207.

19. Pant, B. D., Kumar, S., & Chauhan, S. (2010). Common fixed point of weakly compatible maps on intuitionistic fuzzy metric spaces. Journal of Advanced Studies in Topology, 1(1), 41-49.

20. Demir, I. (2021). Fixed Point Theorems in complex valued fuzzy b-metric spaces with application to integral equations. Miskolc Mathematical Notes, 22(1), 153-171.

21. Manro, S., Kumar, S., Bhatia, S. S., & Tas, K. (2013). Common fixed point theorems in modified intuitionistic fuzzy metric spaces. Journal of Applied Mathematics, 2013(1), 189321.

22. Huang, H., Carić, B., Došenović, T., Rakić, D., & Brdar, M. (2021). Fixed-point theorems in fuzzy metric spaces via fuzzy F-contraction. Mathematics, 9(6), 641.

23. Roldán-López-de-Hierro, A. F., de la Sen, M., Martínez-Moreno, J., & Roldán-López-de-Hierro, C. (2015). An approach version of fuzzy metric spaces including an ad hoc fixed point theorem. Fixed Point Theory and Applications, 2015, 1-23.

24. Abu-Donia, H. M. (2007). Common fixed point theorems for fuzzy mappings in metric space under ϕ-contraction condition. Chaos, Solitons & Fractals, 34(2), 538-543.

25. Plebaniak, R. (2014). New generalized fuzzy metrics and fixed point theorem in fuzzy metric space. Fixed Point Theory and Applications, 2014, 1-17.

26. Roldán-López-de-Hierro, A. F., & Sintunavarat, W. (2016). Common fixed point theorems in fuzzy metric spaces using the CLRg property. Fuzzy Sets and Systems, 282, 131-142.

27. Razani, A. (2005). A contraction theorem in fuzzy metric spaces. Fixed Point Theory and Applications, 2005, 1-9.

28. Adibi, H., Cho, Y. J., O‘Regan, D., & Saadati, R. (2006). Common fixed point theorems in L-fuzzy metric spaces. Applied Mathematics and Computation, 182(1), 820-828.

29. Zhou, M., Saleem, N., Liu, X., Fulga, A., & Roldán López de Hierro, A. F. (2021). A new approach to Proinov-type fixed-point results in non-Archimedean fuzzy metric spaces. Mathematics, 9(23), 3001.

30. Arora, R., & Kumar, M. (2016). Unique fixed point theorems for α–ψ-contractive type mappings in fuzzy metric space. Cogent Mathematics, 3(1), 1183286.